Let the medians of $\triangle ABC$ through $A$, $B$, and $C$ intersect the circumcircle of $\triangle ABC$ again at $D$, $E$, and $F$, respectively. Given $D$, $E$, and $F$, construct $\triangle ABC$.
I'm not sure how to start this problem. Medians are sort of nasty from a circumcircle perspective in my experience, so I've been trying to take advantage of the millions of "construction-related" synthetic properties of their isogonal conjugates, symmedians (namely tangents, cross-ratios which are easily created using inversion, etc.).
It's also immediate that we have the circumcircle (and hence the circumcenter) of $\triangle ABC$, but making use of the relationships between the circumcenter and the medians via, say, the perpendicular bisectors is also tricky since obviously I don't have the sides of the triangle.
Something else I've tried is letting $X$, $Y$, and $Z$ denote the intersections of the symmedians through $A$, $B$, and $C$ with the circumcircle of $\triangle ABC$, respectively, and noting that the symmedian of $\triangle XYZ$ equals the symmedian point of $\triangle ABC$. But no such relationships seem to be present with $\triangle DEF$. Taking tangents at $D$, $E$, and $F$ don't seem to help, either.
There's also the fact that if we project from $A$ the point at infinity on $BC$, $B$, $C$, and the midpoint of $BC$ onto the circumcircle, we have a harmonic quadrilateral. In other words, reflecting $A$ across the perpendicular bisector of $BC$ gives us a harmonic quadrilateral with $B$, $D$, and $C$. But I'm not sure how to make use of this either.
In short: I have no idea how to start this problem.
I'm also interested in solving this problem at least partially on my own, so hints or full constructions are welcome, and constructions with little to no motivation also work (since I then have to figure out why they work). Fully synthetic solutions are preferable but feel free to share other methods too.
From this file we know that the centroid of $\triangle ABC$ is one of the two foci $G_1$ and $G_2$ of the Steiner inellipse of $\triangle DEF$. Therefore all you need is to find these points and draw the circumcevian triangles associate with $G_1$ and $G_2$ (with respect to the circumcircle of $\triangle DEF$). Both of these are the only solutions to your problem.
You can find the foci of the Steiner inellipse $\mathcal E$ knowing that its center is the centroid $G$ of $\triangle DEF$; that its major axis has the direction of the internal angle bissector of $\angle F_+GF_-$ in which $F_+$ and $F_-$ are the first and second Fermat points of $\triangle DEF$ (easily constructable with ruler and compass) and finally that the lengths of the major and minor axis of $\mathcal E$ are given by $|GF_+| \pm |GF_-|$. Thus we can find all its 4 vertices since we know the semiaxis are the bissectors of $\angle F_+GF_-$.
There may be an easier way, but this does it. Look: